Make a <div> square when there is a dynamically changing width based on percentage

There are 2 main techniques to keep the aspect ratio of a responsive element, using padding and vw units :
(for a complete solution for a responsive grid of squares, you can see this answer)

Using vw units

You can use vw units to make your elements square and responsive (viewport units on MDN).
1vw = 1% of viewport width so you can set the height of the elements according to the width of the viewport (or height with vh units).
Example with a 4x4 grid :

body{
  margin:0;
  display:flex;
  flex-wrap:wrap;
  justify-content:space-around;
}

div{
    width:23vw; height:23vw;
    margin:1vw 0;
    background:gold;  
}

<div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div>

The same behaviour can be achieved sizing the element accoring to the viewport height usin vh units.

Using padding

Padding is calculated according to the containers width so you can use it to set the height of block according to it's width.
Example with a 4x4 grid :

.wrap {
    width:80%;
    margin:0 auto;
}
.wrap div {
    width:23%;
    padding-bottom:23%;
    margin:1%;
    float:left;
    background:gold;
}

<div class="wrap">
    <div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div>
</div>

To make a div a square is pretty easy with CSS. You set a width, let's say 50%. Then you add a padding-bottom of the same value:

div {
width: 50%;
padding-bottom: 50%;
}

and it will stay square whenever you resize the window.

.

.

You can do this with any side ratio you want, if you want the box to be 16:9 you calculate:

9/16= 0.56

which you then multiply by the width of your element (in this case 50%(=0.5)), so:

9/16*0.5= 0.28 = 28%

This is un-tested, I do not know of how to do this in CSS only, I would use jQuery.

$('div').height($('div').width());